Mahler’s first conjecture (symmetric case)

Prove that for every origin-symmetric convex body K ⊂ R^n, the Mahler volume M(K) = n! |K| |K°| satisfies M(K) ≥ 4^n, with equality attained by both the Euclidean ball B_2^n and the cube B = [-1,1]^n.

Background

For a convex body K ⊂ R^n, the Mahler volume is M(K) = n! |K| |K°|, which is invariant under GL(n,R). Bourgain–Milman established a bound of the form M(K) ≥ c^n for some universal c > 0, but the sharp constant is unknown.

Mahler’s first conjecture predicts the optimal constant c = 4 in the symmetric case, with extremals given by the Euclidean ball and the cube.